这部分需要会什么？
- 解一元一次不等式并表示解集
- 识别端点是否包含与边界线是否实线
- 检验二元线性不等式的公共可行域
核心概念与公式
不等式两边加减同一个数，不等号方向不变；乘除正数方向不变，乘除负数必须反向。例如 \(-2x>6\) 等价于 \(x<-3\)。复合不等式的“且”表示交集，“或”表示并集。
二元不等式先画边界直线，再用不在边界上的测试点判断哪侧满足。\(\le\)、\(\ge\) 包含边界，使用实线；\(<\)、\(>\) 不含边界，使用虚线。可行域需同时满足全部条件。
做题时怎样使用
- 将文字中的至少、至多、超过分别转成对应不等号
- 解不等式；涉及负数乘除时特别检查方向
- 若变量表示人数、件数等，再在解集中选整数，并核对端点
容易混淆的地方
“至少 5”包括 5；“大于 5”不包括 5。图上的某点满足一个条件，并不代表它满足整个不等式组。
用一个例题串起方法
知识点教学例题一项活动收取固定费用 18 元，每位参加者另付 7 元。总预算不超过 60 元，最多可有多少位参加者？
展开解题思路 ＋
设人数为非负整数 \(n\)，则 \(18+7n\le60\)，所以 \(n\le6\)。最大人数为 6，此时费用恰好为 60 元。
本知识点题库
先独立作答，再查看答案与解析。
线性不等式与可行域挑战
y > 9x -5
For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?
原库参考解析
线性不等式与可行域进阶
A small business owner budgets $2,400 to purchase books each year. The owner must purchase a minimum of 225 books for the year to maintain the discounted pricing. If the owner pays $4.20 per book to purchase paperback books and $11.60 per book to purchase hardcover books, what is the maximum number of hardcover books the owner can purchase to stay within the budget and maintain the discounted pricing?
原库参考解析
线性不等式与可行域进阶
In the xy-plane, which of the following does NOT contain any points (x, y) that are solutions to 7x+4y>12?
原库参考解析
线性不等式与可行域进阶
ln December 2017, the lowest temperature recorded in a certain city was 40 degrees Fahrenheit (°F) and the highest temperature recorded was 90 °F. Which inequality is true for all values of t, where t represents any temperature, in °F, recorded in the city in December 2017?
原库参考解析
线性不等式与可行域进阶
In December 2017, the lowest temperature recorded in a certain city was 40 degrees Fahrenheit (°F) and the highest temperature recorded was 90°F. Which inequality is true for all values of t, where t represents any temperature, in °F, recorded in the city in December 2017?
原库参考解析
线性不等式与可行域进阶
A number x is at least twice the value of a number y. If the value of y is 65, which inequality represents the possible values of x?
原库参考解析
线性不等式与可行域进阶
A number x is less than 2 more than \( \frac{1}{7} \) times the value of a number y. If y is an integer and x = 9, what is the least possible value of y?
原库参考解析
线性不等式与可行域挑战
In the xy-plane, which of the following does NOT contain any points that are part of the solution set to \( 3x−7y>21 \) ?
原库参考解析
线性不等式与可行域进阶
A model estimates that a gray whale travels 73 to 78 miles each day during its migration. Based on this model, which inequality represents the estimated total number of miles, x, a gray whale could travel in 48 days of its migration?
原库参考解析
线性不等式与可行域挑战
\( y<−6x−16 \)
\( y>−4x−10 \)
For which of the following tables are all the values of x and their corresponding values of y solutions to the given system of inequalities?
原库参考解析
线性不等式与可行域进阶
A biologist is designing a study to observe the behavior of male and female amethyst-throated hummingbirds. According to the study's design, at least 91 hummingbirds will be observed, and the positive difference between the number of males, x, and the number females, y, will not exceed 8. Which of the following systems of inequalities represents this situation?
原库参考解析
线性不等式与可行域进阶
The shaded region shown in the graph represents all the solutions to which inequality?
原库参考解析
